Schr[itex]\ddot{\text{o}}[/itex]dinger and Heisenberg's mechanics are the same mechanics.

Dirac showed this where in the Schr[itex]\ddot{\text{o}}[/itex]dinger picture, the states evolve in time, and the observables are stationary, while in the Heisenberg picture, the states are stationary, and the observables evolve in time.

Every calculation gives you the exact same result in both pictures, though one picture might be easier than the other to work with, depending on the problem at hand.

The reason Heisenberg's uncertainty relation is obeyed by electrons is because they exhibit wave-particle duality, just as all quantum objects do. The probability distributions that orbitals represent are not those of classical ignorance, but true quantum uncertainty. It's not true that electrons are well defined and we just can't in principle make up an experiment to show us exactly where one is and where it's going (these sorts of interpretations are ruled out by violating Bell inequalities).

The classical perspective of an electron becomes especially hard to maintain with electrons in P-orbitals which have a nodal plane where the probability is precisely zero. The electrons somehow can still go from one side to the other of this nodal plane even though they cannot strictly speaking, pass through.

When dealing with problems like this it's sometimes best to consider these objects (electrons, photons, etc) as a class of their own, because we just don't experience these sort of phenomena in the everyday macroscopic world.

It's not true that electrons are well defined and we just can't in principle make up an experiment to show us exactly where one is and where it's going (these sorts of interpretations are ruled out by violating Bell inequalities).

The classical perspective of an electron becomes especially hard to maintain with electrons in P-orbitals which have a nodal plane where the probability is precisely zero. The electrons somehow can still go from one side to the other of this nodal plane even though they cannot strictly speaking, pass through.

To be clear, electrons are always described by their wavefunctions, which are abstract "probability distributions", but they can be localized, such as the instance of an electron striking a detector. In this case, the wave function is an eigenstate of the position operator and is described the by delta distribution. The "wave/particle duality" thing is kind of a silly idea since 99.9999% of the time when people are actually discussing such things it's always in terms of waves. Please see the following reference for more details:
Quantum mechanics: Myths and facts http://xxx.lanl.gov/abs/quant-ph/0609163

Also, the business with the nodes is often overstated. For one, the one electron wave functions don't describe an electron "moving" anywhere, they're a static entity that describe the probability of an electron's position. In addition, the probability at the nodes isn't identically zero in the most accurate (relativistic) treatment. Please see the very neat article:
Relativistic quantum chemistry: The electrons and the nodes
J. Chem. Educ., 1968, 45 (9), p 558

For me it's a useful way of explaining that quantum objects are in a class of their own.

When dealing with position and momentum, we do describe quantum systems with "wave functions", but with other observables like spin and polarization, we have a more general object called a state vector which gives the probability amplitudes for outcomes of a given experiment.

That nodal planes don't exist in the full relativistic treatment of the electron is news to me, so thanks for the article (which was indeed a good read). I can imagine it makes sense in that if your probability amplitude is a complex valued function, it doesn't have to pass through zero to go from a positive to a negative value.